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arXiv:0910.5759

Secure Source Coding with a Helper

Ravi Tandon, Şennur Ulukuş, Kannan Ramchandran

Published Oct 30, 2009• 3 citations
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Abstract

We consider a secure source coding problem with a rate-limited helper. In particular, Alice observes an independent and identically distributed (i.i.d.) source X and wishes to transmit this source losslessly to Bob over a rate-limited link. A helper (Helen), observes an i.i.d. correlated source Y and can transmit information to Bob over a separate rate-limited link. A passive eavesdropper (Eve) can observe the coded output of Alice, i.e., the link from Alice to Bob is public. The uncertainty about the source X at Eve, is measured by the conditional entropy of the source given the coded output of Alice. We completely characterize the rate-equivocation region for this secure source coding model, where we show that Slepian-Wolf binning of X with respect to the coded side information received at Bob is optimal. We next consider a modification of this model in which Alice also has access to the coded output of Helen. For the two-sided helper model, we characterize the rate-equivocation region. While the availability of side information at Alice does not reduce the rate of transmission from Alice, it significantly enhances the resulting equivocation at Eve. In particular, the resulting equivocation for the two-sided helper case is shown to be min(H(X),R_y), i.e., one bit from the two-sided helper provides one bit of uncertainty at Eve. From this result, we infer that Slepian-Wolf binning of X is suboptimal and one can further decrease the information leakage to the eavesdropper by utilizing the side information at Alice. We finally generalize these results to the case in which there is additional un-coded side information W available at Bob and characterize the rate-equivocation regions under the assumption that Y-X-W forms a Markov chain.

DOI
10.48550/arxiv.0910.5759

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We consider a secure source coding problem with a rate-limited helper. In particular, Alice observes an independent and identically distributed (i.i.d.) source X and wishes to transmit this source losslessly to Bob over a rate-limited link. A helper (Helen), observes an i.i.d. correlated source Y and can transmit information to Bob over a separate rate-limited link. A passive eavesdropper (Eve) can observe the coded output of Alice, i.e., the link from Alice to Bob is public. The uncertainty about the source X at Eve, is measured by the conditional entropy of the source given the coded output of Alice. We completely characterize the rate-equivocation region for this secure source coding model, where we show that Slepian-Wolf binning of X with respect to the coded side information received at Bob is optimal. We next consider a modification of this model in which Alice also has access to the coded output of Helen. For the two-sided helper model, we characterize the rate-equivocation region. While the availability of side information at Alice does not reduce the rate of transmission from Alice, it significantly enhances the resulting equivocation at Eve. In particular, the resulting equivocation for the two-sided helper case is shown to be min(H(X),R_y), i.e., one bit from the two-sided helper provides one bit of uncertainty at Eve. From this result, we infer that Slepian-Wolf binning of X is suboptimal and one can further decrease the information leakage to the eavesdropper by utilizing the side information at Alice. We finally generalize these results to the case in which there is additional un-coded side information W available at Bob and characterize the rate-equivocation regions under the assumption that Y-X-W forms a Markov chain.
We consider a secure lossless source coding problem with a rate-limited helper. In particular, Alice observes an independent and identically distributed (i.i.d.) sourceXnand wishes to transmit this source losslessly to Bob over a rate-limited link of capacity not exceedingRx. A helper, say Helen, observes an i.i.d. correlated sourceYnand can transmit information to Bob over another link of capacity not exceedingRy. A passive eavesdropper (say Eve) can observe the coded output of Alice, i.e., the link from Alice to Bob is public. The uncertainty about the sourceXnat Eve (denoted by Δ) is measured by the conditional entropy [(H(Xn|Jx))/(n)] , whereJxis the coded output of Alice andnis the block length. We completely characterize the rate-equivocation region for this secure source coding model, where we show that Slepian-Wolf binning ofXnwith respect to the coded side information received at Bob is optimal. We next consider a modification of this model in which Alice also has access to the coded output of Helen. We call this model as the two-sided helper model. For the two-sided helper model, we characterize the rate-equivocation region. While the availability of side information at Alice does not reduce the rate of transmission from Alice, it significantly enhances the resulting equivocation at Eve. In particular, the resulting equivocation for the two-sided helper case is shown to be min(H(X),Ry), i.e., one bit from the two-sided helper provides one bit of uncertainty at Eve. From this result, we infer that Slepian-Wolf binning ofXis suboptimal and one can further decrease the information leakage to the eavesdropper by utilizing the side information at Alice. We, finally, generalize both of these results to the case in which there is additional uncoded side informationWnavailable at Bob and characterize the rate-equivocation regions under the assumption thatYn→Xn→Wnforms a Markov chain.
We consider a secure lossless source coding problem with a rate-limited helper. In particular, Alice observes an i.i.d. source Xnand wishes to transmit this source loss-lessly to Bob at a rate Rx. A helper, say Helen, observes a correlated source Ynand transmits at a rate Ryto Bob. A passive eavesdropper can observe the coded output of Alice. The equivocation ¿ is measured by the conditional entropy H(Xn\ Jx)/n, where Jxis the coded output of Alice. We first completely characterize the rate-equivocation region for this secure source coding model, where we show that Slepian-Wolf type coding is optimal. We next study two generalizations of this model and provide single-letter characterizations for the respective rate-equivocation regions. In particular, we first consider the case of a two-sided helper where Alice also has access to the coded output of Helen. We show that for this case, Slepian-Wolf type coding is suboptimal and one can further decrease the information leakage to the eavesdropper by utilizing the side-information at Alice. We finally generalize this result to the case when there are both secure and insecure rate-limited links from Helen and additional uncoded side informations Wnand Znavailable at Bob and Eve, respectively. For this model, we provide a complete characterization of the rate-equivocation region when Yn¿ Xn¿ (Wn, Zn) forms a Markov chain.
The problem of secure multiterminal source coding with side information at the eavesdropper is investigated. This scenario consists of a main encoder (referred to as Alice) that wishes to compress a single source but simultaneously satisfying the desired requirements on the distortion level at a legitimate receiver (referred to as Bob) and the equivocation rate—average uncertainty—at an eavesdropper (referred to as Eve). It is further assumed the presence of a (public) rate-limited link between Alice and Bob. In this setting, Eve perfectly observes the information bits sent by Alice to Bob and has also access to a correlated source which can be used as side information. A second encoder (referred to as Charlie) helps Bob in estimating Alice's source by sending a compressed version of its own correlated observation via a (private) rate-limited link, which is only observed by Bob. For instance, the problem at hands can be seen as the unification between the Berger–Tung and the secure source coding setups. Inner and outer bounds on the so-called rate-distortion-equivocation region are derived. The inner region turns to be tight for two cases: 1) uncoded side information at Bob and 2) lossless reconstruction of both sources at Bob—secure distributed lossless compression. Application examples to secure lossy source coding of Gaussian and binary sources in the presence of Gaussian and binary/ternary (respectively) side informations are also considered. Optimal coding schemes are characterized for some cases of interest where the statistical differences between the side information at the decoders and the presence of a nonzero distortion at Bob can be fully exploited to guarantee secrecy.
We investigate the problem of secure source coding with a two-sided helper in a game-theoretic framework. Alice (A) and Helen (H) view iid correlated information sequences Xnand Ynrespectively. Alice communicates to Bob (B) at rate R, while H broadcasts a message to both A and B at rate RH. Additionally, A and B share secret key K at rate R0that is independent of (Xn, Yn). An active adversary, Eve (E) sees all communication links while having access to a (possibly degraded) version of the past information. We characterize the rate-payoff region for this problem. We also solve the problem when the link from A to B is private. Our work recovers previous results of Schieler-Cuff and Kittichokechai et al.
In the secure cooperative source-coding problem that we consider, Encoder 1 and Decoder 1 observe, respectively, a pair of independent identically distributed (i.i.d.) correlated sources and wish to communicate the first source to Decoder 2 subject to a distortion constraint. Encoder 1 sends a message to both the decoders over a public nonsecure channel, and then Decoder 1-the helper-sends a message just to Decoder 2 over a secure channel. An eavesdropper, which has access to memoryless side-information that is correlated with the first source, intercepts the message sent by Encoder 1 and based on both wishes to minimize its uncertainty about the first source. The secrecy measure of the system is the equivocation of the first source at the eavesdropper. We determine the rates equivocation region for this problem assuming that conditioned on the side-information of Decoder 1 the eavesdropper's side-information is independent of the first source. To that end, we derive an inner and an outer bound that depend explicitly on the proportional rate that Encoder 1 spends in the cooperation process. Subsequently, we evaluate this region for the case of a doubly symmetric binary source and Hamming distortion.
New inner and outer bounds on the achievable compression–equivocation rate region for generalised secure data compression with side information are given that do not match in general. In this setup, two senders, Alice and Charlie intend to transmit information to Bob via channels with limited capacity so that he can reliably reconstruct their observations. The eavesdropper, Eve, has access to one of the channels at each instant and is interested in the source of the same channel at the time. Bob and Eve also have their own observations, which are correlated with Alice's and Charlie's observations. In this model, two equivocation and compression rates are defined with respect to the sources of Alice and Charlie. Furthermore, different special cases are discussed where the inner and outer bounds match. Our model covers the previously obtained results as well.